Quantum mechanics and geodesic deviation in the brane world

We investigate the induced geodesic deviation equations in the brane world models, in which all the matter forces except gravity are confined on the 3-brane. Also, the Newtonian limit of induced geodesic deviation equation is studied. We show that in the first Randall-Sundrum model the Bohr-Sommerfeld quantization rule is as a result of consistency between the geodesic and geodesic deviation equations. This indicates that the path of test particle is made up of integral multiples of a fundamental Compton-type unit of length $h/mc$.

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